Q.Lilac branches falling off
When we bought our home 14 years ago, there were 2 well established lilac bushes on the border of the yard. Over the years the bushes have begun to look a little haggard. This past spring, we cut them back to about 36″ tall. New growth has sprung from the stumps; however, it seems that new branches are randomly dropping off of the original stumps. What is wrong? Are our bushed beyond saving?

This can happen on old growth of the tree base.
My own aged Lilac has lost a few branches from the older portion of the base of the plant.
Here is a link with more information.
https://www.gardeningknowhow.com/ornamental/shrubs/lilac/pruning-lilac-bushes-when-to-trim-lilac-bushes.htm
https://www.gardeningknowhow.com/ornamental/shrubs/lilac/lilac-care-growing-and-planting-lilac-bush-plants.htm